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Sparse representation techniques have become an important tool in image processing in recent years, for coding, de-noising and in-painting purposes, for instance. They generally rely on an lscr2-lscr1 penalized criterion and fast algorithms have been proposed to speed up the applications. We propose to replace the lscr2-part of the criterion, which has been chosen both for its easy implementation and its relation to the PSNR quality measure, by a lscr∞-part. We present a new fast way to minimize a lscr∞- lscr1 penalized criterion and assess its potential benefits for image De-noising and coding.